To describe the optimal early exercise region at date \( t_{k},\, k=1,\ldots , K, \)  one can divide \( \mathbb {R}^{2} \)  into three different connected sets: one exercise region and two continuation regions (see [1]} for more details).
All these regions can be parameterized by using two functions depending on two dimensional parameter
\( \theta _{k}\in \mathbb {R}^{2}. \)  Making use of this characterization, we define a parametric family of stopping regions as in [2]} via
\(\mathcal {S}_{k}(\theta _{k}):=\lbrace (x_{1},x_{2}): \max (\max (x_{1},x_{2})-K,0)>\theta ^{1}_{k};\, |x_{1}-x_{2}|>\theta ^{2}_{k} \rbrace ,\) 
